We could rationalize the numerical results by examining the flow in the narrow zone where the liquid / liquid and liquid / air interfaces get very close, which we dubbed virtual contact point.
We showed that the shear stresses at the plate and at the liquid / liquid interface at that point alone are sufficient to predict the thicknesses of the two films, through simple scaling laws derived from the original ideas of [1]}.
Regarding the influence of the different control parameters of the problem, we found that the thickness \(\hat{h}_{1,\infty }\)  of the lowermost film is barely affected by the viscosity ratio \(M\) .
This is a consequence of the shear stress at the interface separating the two liquids being negligible as compared to the one at the plate / lower liquid interface.
On the contrary, the thickness of the floating layer, \(\Delta H\) , has a strong impact on \(\hat{h}_{1,\infty }\) , which grows monotonically with \(\Delta H\) .
In this two-liquid configuration, the thickness of the lower film is always larger than the corresponding thickness for a one-liquid Landau-Levich flow.
The thickness \(\delta \hat{h}_\infty \)  of the uppermost film exhibits a comparatively more complex behaviour, showing a non-monotonous trend with both \(M\)  and \(\Delta H\) , with a maximum for a given pair of these parameters.
More importantly, there is a finite range of values in the (\(M\) , \(\Delta H\) ) parameter space where \(\delta \hat{h}_\infty \)  takes physically-realizable values, which amounts to say, where a solution with two entrained films exists.
